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dent, the components of r along any one of the axes must equal the sum of the corresponding components of the given vectors. Therefore (3) can be split into the following three separate equations.
rtf = ay+btf+ctf +  -  ,                         (4)
rz = a, + b, + c, + -..
It was shown in  11 that when two vectors are parallel the algebraic sum of their magnitudes equals the magni-
Fro. 12.
tude of their resultant. This result may be extended to any number of parallel vectors. Therefore we can put the vector equations of (4) into the following algebraic forms:
rv- av+lv+cy+
Equations (5) determine r through the following relations:
(6) (7)
9"  rrr
COS oti =
COS C*2 ^ ........J
cos 3=
where i, 2, and 3 are the angles r makes with the axes.
16. Multiplication and Division of a Vector by a Scalar. When a vector is multiplied or divided by a scalar the result is a vector which has the same direction as the original vector. If, in the equation b = ma, m be a scalar then b has the same direction as a but its magnitude is m times that of a.